This is a topic I've mentioned a few times before. Essentially the structure of matter in quantum gauge field theories is unclear to me. I have no clear question here, just some initial discussion points.

So at the first level, it seems a particle based view of quantum field theory is difficult to maintain. In brief on the Hilbert Spaces of interacting field theories there is no well-defined number operator ##N##. In some texts it's said that states in the interacting theory are a superposition of different particle numbers, but in reality the fact that an ##N## operator doesn't exist means there are no states with a well-defined particle number. You can see this explicitly in Glimm, Osterwalder and Schrader's work on ##\phi^{4}_{3}##, where the number operator for the interacting theory diverges as the cutoff is removed.
In addition to this we have Malament's theorem where he shows that relaitivistic quantum theories don't possess any states with the intuitive property of particles. In essence he shows that there are no local particle creation operators.

Then we have the usual picture of fields and particles being their excitations. Now if particle like states aren't elements of interacting field theory Hilbert spaces, I don't think one can say they are excitations of fields. What one can say is that most field theories can be shown to give rise to states that at late times can be proven to have a well defined number of "clicks" they cause in detectors, with the detectors being represented by specific local observables (details are to be found in the monographs of Araki and Haag, which we can go into, but I don't want to overload the initial post).

At this point we might turn to fields as the fundamental objects. However this remains tricky to me, let's look at gauge theories.

The problem with Gauge theories is that fields carrying the gauge charges cannot be local. If ##\Omega_{\Delta}## is a projector onto states confined to the region ##\Delta## of Minkowski spacetime and ##A## is an observable carrying the gauge charge, then it is always the case that ##\Omega_{\Delta}A \neq 0##. For this reason parameters like ##x## in the quark field ##\psi(x)## are formal, carried over from classical notation, but don't reflect actual localisation at a point.

If one wants ##\psi(x)## to be local, it can only be so as an operator on a Hilbert-Krein space, not on a Hilbert Space of physical states. If one does this there needs to be a condition selecting out the subset of the Hilbert-Krein space containing the physical states (BRST condition in non-rigorous language).

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".

So in something like QED the electron and photon fields aren't well-defined local operators and thus actual physical states (which can be localised) can't be associated with them directly. In QCD it is even worse, where the quark and gluon fields, even ignoring this problem, would map one out of the physical Hilbert Space regardless as they carry color charge when all physical states are colorless.

Gauge symmetry itself is formulated in terms of these formal objects ##\psi(x), A_{\mu}(x)## carrying guage charges and most properties of such gauge symmetries (e.g. conserved charges) do not survive quantization. The only aspect of them that does are local Gauss laws (see Nakanishi "Covariant Operator Formalism of Gauge Theories of Quantum Gravity" and Strocchi's book).

So really QCD's physical content to me appears to be there are states which at early/late times make detectors click in a fashion that corresponds to the properties we give nucleons, i.e. late time states will act like particles with nucleon properties. There is a relation between the scattering angles and amplitudes of these nucleon like states that can be encoded in differential forms expressing local Gauss laws.

However the entire structure of quarks, gluons and to some degree even their local fields, seems like a crutch we use so that we have objects that give nice integrals and implement the Gauss law in a way that is easy to use (as Guage "charges") since it reduces it to group theoretic calculations. However the price is that these objects are a long way from the physical content of the theory.

This seems to render statements like "The proton is made of three quarks" into shorthand for "There is a state which at late times has localized properties x,y,z. The restriction the Gauss law imposes on its amplitudes can be modelled perturbatively by decomposing its creation operator into a product of three fields containing charges. Though the states corresponding to these fields are unphysical, lacking even positivity, we have conditions (BRST) to recover what we need for the late time particulate state"

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".

Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846

In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.

There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?

A direct coupling between the field strengths and the electron field will either give chargeless electron fields or be nonlocal.

Certainly it can be done, but it would be more in terms of a nucleon Lagrangian in QCD's case having some sort of Gauss's law encoded in a differential form which is some function of the Nucleon fields.

A far cry from the fields that appear in the usual Lagragian. I doubt the physicality of the fields we typically use, not fields in general.

There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?

I believe that, just like particle numbers, whatever couplings are ''introduced'' by a Lagrangian ansatz, these are washed away by renormalization, hence are not rigorously meaningful notions. However the local fields that survive renormalization in a rigorous limit are of course correlated, unlike their free counterparts. This implies interaction.

In electrodynamics one does not need a notion of local charge but only those of a 4D charge current and an electromagnetic field, as these are the only fields that figure classically. Thus the vacuum sector of QED describes everything needed to recover macroscopic electrodynamics. Charged states cannot be found there, of course, but these can be taken to be an unphysical idealization. Rays of charged particles may be viewed as approximate notions that can probably be modelled by appropriate charge current distributions. I haven't seen this done but I don't see any obstacle in doing this.

I understand much less about QCD, but there the matter content should be describable too by local and gauge invariant quark currents. Again the vacuum sector should describe everything of true physical relevance. For example, current-current correlations are among the observables that reveal information testable by experiment. See, e.g., the book ''Quantum chromodynamics: an introduction to the theory of quarks and gluons'' by Yndurain.

The currents exist (I'll have more to say on them later) and are local fields, but their decomposition or expression in terms of quark fields is not directly physically sensical, as the quark fields only exist on an enlarged Hilbert-Krein space.

This to me obscures the physical content of the standard description. Really we should say there are nucleon currents obeying Gauss laws. Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents and if we had full nonperturbative control of the theory quarks could be eliminated.

I'm saying something similar to what you say about virtual particles I guess. They're a concept useful for calculations, but not part of the theory's genuine physical content. Same with gluons.

Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents

I agree that quark fields themselves are unphysical. But what is your argument that quark currents (the renormalized version of the corr. quadratics in the quark fields) cannot be physical? They should exists in the 6 quark flavors (or, ignoring spin, as a 6x6 matrix valued field), and hence should deserve to be called quark currents, even though the quarks themselves (as fields or particles) are virtual only.

I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.

I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.

Would this seem reasonable to you?

I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.

But there are far too many baryons, and the flavor information is lost.

Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.

In particular in section 5.3 they discuss the physical content of QCD. Now their discussion has certain propositions they can't prove (if they could it would constitute a rigorous construction of Yang-Mills), but under these assumptions the Wightman axioms imply:
$$\mathcal{H}_{phys} = \overline{\mathcal{A}\left(\mathcal{O}\right)\Omega}$$
That is the physical Hilbert space can be constructed from the closure of local hadron fields operating on the vacuum. To be clear ##\mathcal{A}\left(\mathcal{O}\right)## is the local algebra of color singlet local fields, hence this is essentially a Reeh-Schlieder type theorem.

I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.

But there are far too many baryons, and the flavor information is lost.

Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.

I'm a bit puzzled by these statements... Of course the main force to keep "quarks and gluons" bound (even confined!) in hadrons, which are so far the only asymptotic states of QCD that can be measured, is the strong force. The only "ab-initio way" to understand hadrons from QCD are lattice-QCD calculations. These are Monte-Carlo evaluations of appropriate gauge-invariant correlation functions. One of the key achievements of this approach is a pretty accurate calculation of the hadronic mass spectrum of the empirically known as well as not-yet seen hadrons.

I've no clue what you mean by "the flavor information is lost". Of course to get the hadron spectrum the lattice-QCD calculations involve correlation functions with the right quantum numbers, including flavor, i.e., the corresponding valence-quark flavor.

Another approach to understand hadron phenomenology are of course effective low-energy models. The most important approach in the light-quark sector is chiral perturbation theory (and unitarized versions thereof). This uses the approximate chiral symmetry of QCD in the light-quark sector (flavor SU(2) for up and down or flavor SU(3) for up, down, and strange quarks), which is spontaneously broken in the vacuum and at low temperatures and or (net-baryon) densities. These models are governed by the symmetry properties of corresponding composite fields. Also these involve of course well-defined flavor degrees of freedom. For a nice introduction to this approach, see

Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846

In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.

I just read this again and my phrasing was very bad. I meant to say the concept of local field is dubious for the quark and gluon fields and the only versions of them that are well-defined on the physical Hilbert space are loop-like, see Bert Schroer's paper here:https://arxiv.org/abs/1601.04528

To summarise, in QCD the real physical description involves a complicated non-Fock space on which we have only hadron operators.

However this leads into my next point. If quarks fields had been defined on the physical Hilbert Space, they would have formed a basis (in the operator theoretic sense) for all other local fields and thus some sense could be given to the notion that "protons are made of quarks, pions are made of quarks".

However since the only physical fields are hadrons what now is the picture of their composition? They seem fundamental. Perhaps there is a basic set of hadron fields that others can be expressed as functions of, but I don't know if there is a unique choice of such.

You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.

You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.

Sorry, hadron and glueballs then. More so the point is that what in the conventional way of writing the theory are composite fields are in fact fundamental and that this seems to me to cause a funny detail with the matter, i.e. hadron, fields in that without quarks they aren't all reducible to being "composed" of a unique common set of fields.

Do you mean the flavor information becomes obscured or do you mean it is literally not present?

What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.